<%BANNER%>

Numerical Study on Tidal Flow over Hollows in Estuaries

Permanent Link: http://ufdc.ufl.edu/UFE0042362/00001

Material Information

Title: Numerical Study on Tidal Flow over Hollows in Estuaries
Physical Description: 1 online resource (52 p.)
Language: english
Creator: Tian, Miao
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: geometry, headland, hollow, momentum, roms, stratification, topography
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Coastal and Oceanographic Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Observations showed a tidally asymmetry of the flow over a hollow in an estuary. A series of numerical experiments were carried out to simulate the tidal flow in an idealized estuary with a hollow in the channel. The purpose of this study is to explore: a) effects of stratification of the estuary on the flow; b) effects of hollow geometry on the oscillatory flow; c) effects of headland on the flow; and d) the momentum balance for the flow over a hollow in a vertically homogeneous estuary. Results showed that the horizontal density gradient plays an important role in determining the asymmetric distribution of the flow in one tidal cycle. Moreover, elliptical and deeper hollows enhanced the asymmetry than circle and shallow ones. The asymmetric shape of the channel along the hollow alerts the flow by making it converge and diverge when it flows across the headland.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Miao Tian.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Valle-Levinson, Arnoldo.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042362:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042362/00001

Material Information

Title: Numerical Study on Tidal Flow over Hollows in Estuaries
Physical Description: 1 online resource (52 p.)
Language: english
Creator: Tian, Miao
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: geometry, headland, hollow, momentum, roms, stratification, topography
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Coastal and Oceanographic Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Observations showed a tidally asymmetry of the flow over a hollow in an estuary. A series of numerical experiments were carried out to simulate the tidal flow in an idealized estuary with a hollow in the channel. The purpose of this study is to explore: a) effects of stratification of the estuary on the flow; b) effects of hollow geometry on the oscillatory flow; c) effects of headland on the flow; and d) the momentum balance for the flow over a hollow in a vertically homogeneous estuary. Results showed that the horizontal density gradient plays an important role in determining the asymmetric distribution of the flow in one tidal cycle. Moreover, elliptical and deeper hollows enhanced the asymmetry than circle and shallow ones. The asymmetric shape of the channel along the hollow alerts the flow by making it converge and diverge when it flows across the headland.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Miao Tian.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Valle-Levinson, Arnoldo.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042362:00001


This item has the following downloads:


Full Text

PAGE 1

1 NUMERICAL STUDY ON TIDAL FLOW OVER HOLLOWS IN ESTUARIES By MIAO TIAN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIRMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVE RSITY OF FLORIDA 20 10

PAGE 2

2 20 10 Miao Tian

PAGE 3

3 To my Mom

PAGE 4

4 ACKNOWLEDGMENTS Completing a m aster is truly a marathon event, and I would not have been able to complete this journey without the aid and support of countless people over the past two year s. I must first express my gratitude towards my advisor, Professor Arnoldo Valle Levinson. H is leadership, support, attention to detail, hard work have set an example I hope to match some day. I would like to thank the many gra duate students I have worked with in Valle Levinson s group: Amy Waterhouse Junwoo Lee Berkay N uvitber Cloe Winant, Sangdon So, Kimberly Arnott I wish to thank Peng Cheng in particular: as my colleague his insights and comments were invaluable over the years, and I look forward to a continuing collaboration with him in the future. I also thank some of my classmates : Andrew Lapetina Corey Ramstad Tracy Martz Tianyi Liu S h i feng Su h Luciano L absalonsen Ilgar Safak, Go Fujita They eac h helped make my time in the m aster pro gram more fun and interesting. Finally, I thank my parents uncle and aunt for instilling in me confidence and a drive for pursuing my m aster

PAGE 5

5 TABLE OF CONTENTS ACKNOWLEDGMEN TS ................................ ................................ ................................ .. 4 LIST OF FIGURES ................................ ................................ ................................ .......... 6 LIST OF ABBREVIATIONS ................................ ................................ ............................. 7 ABSTRACT ................................ ................................ ................................ ..................... 8 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ ...... 9 2 MODEL DESCRIPTION ................................ ................................ .......................... 12 3 D ESCRIPTION OF THE C ONCEPTUAL MODEL AND THE VELOCITY RATIO ... 14 Conceptual Model ................................ ................................ ................................ ... 14 Velocity Ratio ................................ ................................ ................................ .......... 15 4 EFFECTS OF STRATIFICATION ................................ ................................ ........... 16 Vertically Homogeneous Case ................................ ................................ ................ 16 Vertically Stratified Case ................................ ................................ ......................... 17 Comparison of Cases ................................ ................................ ............................. 18 5 EFFECT OF HOLLOW GEOMETRY ................................ ................................ ...... 20 Hollow Shape ................................ ................................ ................................ .......... 20 Hollow Topography ................................ ................................ ................................ 20 6 EFFECTS OF MORPHOLOGY ................................ ................................ .............. 22 7 MOMENTUM BALANCE ................................ ................................ ......................... 25 8 CONCLUSION ................................ ................................ ................................ ........ 50 LIST OF REFERENCES ................................ ................................ ............................... 51 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 52

PAGE 6

6 LIST OF FIGURES Figure Page 2 1 A p lan view of th e estuary ................................ ................................ ..................... 29 2 2 A plan view of the hollow region.. ................................ ................................ .......... 30 2 3 Model stability plots. ................................ ................................ ............................. 31 3 1 Depth averaged flow velocity at the center of t he hollow as a function of time ..... 32 4 1 Results of the vertically homogeneous estuary during flood. ................................ 33 4 2 Results of the vertically homogeneous estuary during ebb. ................................ 34 4 3 Results of the vertically stratified estuary during flood. ................................ ......... 35 4 4 Results of the vertically stratified estuary during ebb. ................................ ........... 36 4 5 Comparison of the results during flood. ................................ ................................ 37 4 6 Comparison of the results during ebb. ................................ ................................ .. 38 5 1 The velocity ratio as a function of W / L ................................ ................................ 40 5 2 Velocity ratio as a f unction of ................................ ................................ ..... 40 6 1 Plan view s of two cases of the channel with headland. ................................ ......... 41 6 2 Comparisons of t he results from two cases during flood. ................................ ...... 42 6 3 Comparisons of the results from two cases during ebb. ................................ ........ 43 6 4 Plan view of the depth averaged velocity vector during flood. .............................. 44 6 5 Plan view of the depth averaged velocity vector ebb. ................................ ........... 45 7 1 Cross secti on averaged momentum terms value as a function of time in one tidal cycle. ................................ ................................ ................................ ........... 46 7 2 Higher ordered cross section averaged momentum terms value as a function of time in one tidal cycle. ................................ ................................ .................... 47 7 3 P lan view of contour plots for the momentum ratio during flood. ........................... 48 7 4 P lan view of contour plots for the momentum ratio dur ing ebb .. ........................... 50

PAGE 7

7 LIST OF ABBREVIATION S Reference water d ensity (kg/m 3 ) Longitudinal density gradient (kg/m 4 ) Longitudinal v elocity in the center of the hollow (m/s) Longitudinal v elocity as the flow enters the hollow (m/s) Velocity Ratio A cceleration due to the gravity ( m / s 2 ) Longitudinal distance (m) D istance from the plat bottom to the static water surface (m) M aximum hollow depth below the undisturbed bottom level (m) Water surface elevation (m) Cross section averaged term in longitudinal momentum equation ( m / s 2 ) C ross section area ( m 2 )

PAGE 8

8 ABSTRACT OF THESIS PRESENTED TO THE GRA DUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLME NT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE NUMERICAL STUDY ON TIDAL FLOW OVER HOLLOWS IN ESTUARIES By Miao Tian December 20 10 Chair: Arnoldo Valle Levinson Major: Coastal and Oceanographic Engineering Observations showed a tidally asymmetry of the flow over a hollow in an estuary. A series of numerical experiments were carried out to simulate the tidal flow in an idealized estuary with a hollow in the channel The purpose of this study is to explore: a) effects of stratification of the estuary on the flow; b) effects of hollow geometry on the oscillatory flow; c) effects of headland on the flow; and d) the momentum balance for the flow over a hollow in a vertically homogeneous estuary. Results showed that the horizontal density gradient plays an important role in determining the asymmetric distribution of the flow in one tidal cycle. The one that has the greater horizontal density gradient shows more prominent flow a symmetry in the hollow. Moreover, elliptical and deeper hollows enhanced the asymmetry than circle and shallow ones. T he asymmetric shape of the channel along the hollow alerts the flow by making it converge and diverge when it flows across the headland. I n addition, the main dynamics of the tidal flow in a vertically homogeneous estuary is among the pressure gradient, the vertical friction and the local acceleration. And the momentum transition between the vertical friction and the longitudinal advection c ould also be observed especially during ebb tides.

PAGE 9

9 CHAPTER 1 INTRODUCTION Scour pits or bathymetric hollows, which could be defined as bathymetric depression s in the ocean floor or in coastal embayments affect the hydrodynamic of local currents and wave patterns. Basic Bernoulli dynamics, represented by the momentum balance between barotropic pressure gradient and advective effects, provides a prediction to the hydrodynamics of a homogeneous flow over a hollow. According to Bernoulli dynamics, the velocit ies are expected to decrease as depth incre ase s. M oreover, the flow would produce a symmetric distribution in hydrodynamics along the hollow. H owever, observation s in the C sapeake Bay (Salas Monreal and Valle Levinson, 2009) and numerical studies (Davies and Brown, 2007 & Cheng and Valle Levinson, 2009) were not in agreement with the results predicted by the Bernoulli dynamics. Both observations and models show that tidal flows would accelerate over the deepest part of the hollow during flood; and decelera te at the center of the hollow during ebb. I n other words, an asymmetric distribution of the current velocities during flood and ebb has been described over the hollow. Bernoulli dynamics predicts deceleration of the velocities during ebb, but fails to exp lain the acceleration of flow during flood. Salas Monreal and Valle Levinson (2009) suggested a convergence/divergence pattern as the dominant mechanism for causing the spatial asymmetry of flow velocities over the hollow. According to this pattern, the fl ow should converge when it enters the hollow and diverge when it leaves it. The convergence pattern explains the acceleration of flow over hollow during flood. N evertheless, it fails to clarify the observed deceleration of flow velocities during ebb (Cheng and Valle Levinson).

PAGE 10

10 Davies and Brown (2007) observed enhancement of flow velocity with field data collected at the mouth of Dyfi Estuary. B ased on a series of numerical experiments for simple, steady, channel flow over schematized hollows (Two Dimension al simulations by Telemac2D), they showed that the variation of depth mean velocity was strongly related to the shape of the elliptical hollow, the ratio between depth of the hollow and the depth of the water outside the hollow, and the bed roughness. W ith the combination of a conceptual model and several numerical experiments with the R egional O cean M odeling S yste m, Cheng and Valle Levinson (2009) pointed out that the existence of a horizontal density gradient plays an important role in causing the spatial asymmetry of flow over the hollow. T hey carried out numerical experiments for steady open channel flow to show the effects of hollow geometry on along channel velocity distribution. Moreover, they prescribed a stratified tidal flow in an idealized estuary A n asymmetry of the flow velocity distribution during one tidal cycle had been observed as it was expected. They also pointed out that coastal morphology would also affect the flow. Cheng and Valle Levinson s work pointed out the main factors affecting steady open channel flow. However, whether their conclusion could be applied to unsteady flow needs to be determined. Valle Levinson and Guo observed the asymmetric tidal flow structure in the channel with weak horizontal density gradient and discovered a transition from frictionally dominated dynamics outside the hollow to advectively dominated dynamics over the pit (Valle Levinson and Guo 2009). The momentum transition needs to be further explored by numerical experiments.

PAGE 11

11 In this study, we are intereste d in oscillatory flow over the hollow, which extends the previous research. Two main issues are investigated. First, it explores the main factors affecting tidal flows over the hollow. Second, it analyzes the momentum budget of tidal flows in a vertically homogeneous estuary. A series of numerical experiments were carried out to determine how the stratification of the channel, the hollow geometry and the headland on the coastline affect tidal flows over a hollow. In addition, the main momentum balance for t idal flows over a hollow in a vertically homogeneous channel is figured out. T his work is organized as follows. Chapter 2 describes the numerical model used and the model setup, followed by Chapter 3 that briefly introduces a conceptual model and the conce pt of velocity ratio that will be applied in the following chapter s. Chapter 4 shows the effects of stratification on flow over the hollow. Chapter 5 explores the effects of different hollow geometr ies on tidal flow s Chapter 6 investigates the modificatio ns effected by a headland on the hydrodynamics over the hollow, followed by analysis of the momentum balance in a vertically homogeneous estuary in Chapter 7 The work is concluded in Chapter 8

PAGE 12

12 CHAPTER 2 MODEL DESCRIPTION In order to verify the existenc e of the asymmetric flow pattern over a hollow, a primitive equation numerical model, ROMS (Regional Ocean Model System, Shchepetkin and McWilliams, 2005) has been applie d to simulate a n oscillatory flow over a n idealized hollow located closely to the mout h of a rectangular estuary. The estuary is 3 8 0 km long and 100 km wide with flat bottom (Fig ure 2 1) The center of the hollow is located at 25 km from the estuary mouth T he hollow has a Gaussian shape with 2 0 km in length and 6 .0 km in wid th. T he water d epth surrounding the hollow is 10 m (Fig ure 2 2) The model has a grid of 200 (along the channel, horizontally) by 80 (across the channel, laterally) by 20 (vertically) cells. I n order to produce a highly resolved computational domain near the hollow, t he along channel grid size decreases the hollow s center, and then increases exponentially from the hollow s center to the estuary s head Similarly, the across channel grid is densely distributed close to the hollow and sparsely away from it. The vertical distribution of density in an estuarine channel is controlled by river discharge at the head of the estuary and the amplitude of the tide at the mouth of it. A freshwater discharge of and a semidiurnal tide are spe cified at the head and the mouth correspondingly. A closed boundary condition is imposed at the northern and southern sides of the channel. T he background water temperature is set to be 15 C throughout the entire domain. I nitially, t he salinity field is s pecified as vertically homogeneous and with a uniformly distributed horizontal salinity gradient along the channel ( 1 .16 10 4 psu/ m) T he Coriolis force has been neglected. A quadratic stress was exerted at the bottom with a bottom drag coefficient of 0.00 25. Vertical diffusivity and viscosity are

PAGE 13

13 computed using the k turbulence closure. T o reach steady state, t he model ran for 6 0 days with a time step of 10 seconds The values of the kinetic energy, potential energy and the total energy on the 55 th day a re nearly identical with th ose of the 60 th day (Fig ure 2 3), which means that the model reaches a steady state when it runs for 60 days.

PAGE 14

14 CHAPTER 3 D ESCRIPTION OF THE CO NCEPTUAL MODEL AND T HE VELOCITY RATIO Conceptual Model The impact of the horizontal ba roclinic pressure gradient on the flow speed may be illustrated with a steady, one dimensional equation, which contains the momentum balance between advection and pressure gradient terms (Cheng and Valle Levinson 2009): (3 1) where is the flow velocity in the hollow, is the velocity as the flow enters the hollow, is the acceleration due to the gravity, is the water surface elevation, is a reference water density, is the horizontal direction, is the horizontal density gradient. is the length (horizontal distance) of the hollow is the distance from th e plat bottom to the static water surface, is the maximum hollow depth below the undisturbed bottom level The sign of plays a crucial role in the relation above. During flood, density decreases in the upstream direction (from estuarine mouth to head), which produce s a negative density gradient, i.e. a negative value for Hence the velocity will increase as the water flows over the deepest part of the hollow because the third and fourth term s on the right hand side of Eq. (3 1) cause an increase in Conversely, the

PAGE 15

15 density gradient ha s a positive sign during ebb, and then the velocity will decrease over the hollow (Cheng and Valle Levinson 2009). Moreover, the modifications of the flow as it enters the center of the hollow also depends upon two non dimensional parameters for flow properties: and where is the width (lateral distance respectively) of the hollow It means that t he flow modifications over a channel with a simple hollow of sinusoidal shape are intimately linked to the geometry of th e hollow. Velocity Ratio In this study, we have to compare the velocities of different cases with various hollow geometries and different types of estuaries. A new method was applied to compare the velocity modification over the hollow. Here we introduce t he concept of velocity ratio, which nondimensionalizes the vertically averaged flow velocity in the cross The bottom panel of Fig 2.2 shows the bottom profile (black) along the c enter line of the hollow. We notice that the center of the hollow is located nearly 275 km away from the head of the estuary. We define the depth averaged velocity at this point as During flood, the w ater enters the hollow from right to left. Then we could define the depth averaged velocity before the flow enters the hollow as which is 10 km to the right to the middle. We could quantify the enhancement of the flow velocity when it enters the hollow during flood by calculating Similarly, during ebb, we define the depth averaged velocity before the flow enters the hollow as which is 10 km left to the center. The velocity ratio allows us to compare the variation of the local velocity at the center of the hollow

PAGE 16

16 CHAPTER 4 EFFECTS OF STRATIFIC ATION Two groups of numerical experiments were carried out for this purpose. The model setup is the same as we have described except f or the amplitude of the tide and the amount of freshwater inflow. The stratification of these two groups of experiments is different, as specified by the amplitude of the input tide and the amount of the freshwater. In the first group of experiments, a tid e with amplitude of 1.5 m is prescribed at the mouth of the estuary and a freshwater inflow of 30 is specified on the head to make sure that the estuary is vertically homogeneous. This group contains two experiments: one with the hollow in the channel and one without it. In the second group, the amplitude of the inputting tide is 0.5 meters and freshwater input is 90 The relatively weak tide and strong freshwater inflow create a vertically stratified es tuary. This second group also contains two experiments: with and without the hollow. Vertically Homogeneous Case The case of a vertically homogeneous, but not longitudinally, estuary has been explored here. Figure 4 1 shows the salinity distribution, veloc ity contours and depth averaged velocity profile along the center line of the estuary during flood. Figure 4 2 shows the same plots during ebb. The instantaneous salinity structure shows a vertically homogeneous estuary (upper panel of Fig ures 4 1 and 4 2) The middle panel of Fig ure 4 1 shows a typical snapshot of the flow along the center line of the hollow during flood tides. The depth averaged speed increases from ~25 cm/s on the right edge of the hollow (seaward side of the estuary) to ~45 cm/s over th e left edge of (landward side of the estuary) of the hollow. The enhancement of the velocity seems apparent in the depth averaged velocity profile (bottom panel of Fig ure 4 1). The depth

PAGE 17

17 averaged flow increases roughly linearly over the hollow. It increase s approximately 40% (from 25 cm/s to 35 cm/s) as it moves over the center of the hollow. During ebb tides, the current speed is reduced as the flow entered the landside of the hollow (middle panel of Fig ure 4 2). The minimum speed occurs near the center of the hollow (bottom panel of Fig ure 4 2). This numerical experiment shows an asymmetric structure of the tidal flow over the hollow in a vertically homogeneous estuary. The horizontal density gradient plays an important role in the flow pattern. The model results are in consistence with the observation in the Seto Inland Sea (Valle Levinson and Guo 2009). Vertically Stratified Case The flow over the hollow of a vertically stratified estuary has been studied by Cheng a nd Valle Levinson (2009) (Fig ures 4 3 an d 4 4). During flood, the depth averaged flow velocity increased roughly linearly along the channel. However, the speed increased from ~28 cm/s on the right edge of the hollow (seaward side of the estuary) to ~40 cm/s over the left edge (landward side of t he estuary) when the water goes across the hollow. During ebb tides, the depth averaged flow appears the linearly increased structure along the channel. Converse to the velocity during flood, the current speed was reduced as the flow entered the landside o f the hollow (bottom panel of Fig ure 4 4 ). The results show an asymmetric structure of the tidal flow over the hollow, which agrees with observations in Chesapeake Bay (Cheng and Valle Levinson 2009), although they are obtained for a vertically stratified condition, which means relatively low horizontal density gradient, the results were still in agreement with the conceptual analysis, which indicates the horizontal density gradient is crucial in the hydrodynamics over a hollow.

PAGE 18

18 Comparison o f Cases The de pth averaged velocity ratio and density gradient of the cross section along the main axis of the hollow at the two tidal phases are shown in Fig ure 4 5 During flood the velocity ratios of cases without hollow increase almost linearly from the upstream edg e (seaward side) to the downstream side (landward side). However, we could observe that the ratio of cases with hollow increases more than the case without hollow, and the ratio of the vertically mixed case has more increment than the vertically stratified one ( u pper panel of Fig ure 4 5 ). Moreover, we find that the horizontal density gradient of the vertically homogeneous case is in general greater than that of the vertically stratified one (middle panel of Fig ure 4 5 ). It means that the vertically homogene ous channel has a relatively greater longitudinal baroclinic pressure gradient, which results in a larger velocity ratio in the vertically mixed case. During ebb, the velocity ratio of the cases without the hollow increases slowly seaward. And the velocity ratio of the cases with the hollow reduces as the water goes across the hollow. Moreover, the ratio of the vertically homogeneous case decreases more than that of the vertically stratified one (upper panel of Fig ure 4 6). Similar to the results we obtain ed during flood, the horizontal density gradient seems to be larger in the vertically mixed case than the vertically stratified one (middle panel of Fig ure 4 6). It means that the flow velocity decreases as the water goes across the hollow because of the B ernoulli effects, and the lager baroclinic pressure gradient in the vertically mixed case enhances the reduction in velocity. In these two groups of tests, we find that the vertically homogeneous estuary would have a stronger flow asymmetry than the verti cally stratified estuary because of the existence of larger horizontal density gradient in the hollow region. Moreover, the results

PAGE 19

19 strengthen our understanding that the asymmetric distribution of the baroclinic pressure gradient along the hollow induces t he asymmetry of the flow in one tidal cycle.

PAGE 20

20 CHAPTER 5 EFFECT OF HOLLOW GEO METRY Hollow Shape In order to examine the effect of the hollo w s shape on the oscillatory flow, a group of experiments was carried out. T he model setup for each experiment was the same as described in the vertically homogeneous cases except that each one has different ho llow shape With a fixed hollow length ( L ) and a var ying width ( W ), the ratio between the hollow width and the hollow leng th varies from 0.2 to 1. This group consists of nine cases in total, with the shape of the hollow changing from an ellipse to a circle. The velocity ratio of the depth averaged velocity at the center of the hollow as a function of W/L is shown in Fig ure 5 1. During flood, the velocity ratio increases as W/L increases, indicating that the enhancement of velocity becomes stronger as the hollow tends to be a circle. During ebb, the velocity ratio decreases as W/L in creases, showing that the reduction of veloc ity appears more prominent in a circular than an elliptical hollow. Hollow Topography A group of numerical experiments has been carried out to explore the influence of the hollow topography on the flow. W ith the same model setup as the vertically homogene ous cases except for the hollow depth, this group contains 13 cases, with a fixed surrounding hollow depth which equals to 10m, and various hollow depths T he hollow depth increases 2.5m for each cas e, changing from 0 to 30m. The velocity ratio of the depth averaged velocity at the center of the hollow as a function of is shown in Fig ure 5 2 The velocity ratio increases as increases, while it d ecreases as decreases. The profile clearly indicates that the

PAGE 21

21 increment and decrease caused by the density difference during flood and ebb are more significant in relatively deep hollows than in shallow ones.

PAGE 22

22 CHAPTER 6 EFFECTS OF MORPHOLOG Y Measure ments have show n that the coastal morphology near the hollow c ould affect the flow pattern in the hollow (Davies and Brown 2007). The existence of a headland around the pit could cause convergence as the flow moves over the hollow (Valle Levinson and Guo 200 9 ). However, the relative role played by the morphology compared to the baroclinic pressure gradient needs to be clarified. N umerical experiments are carried out to explore the effects of a headland on the flow p attern in the hollow. Two cases have been tested, which have the same model setup as the vertically homogeneous case except for the shape of the channel. A headland has been added in both cases. In the first case, the headland is locate d near the seaward s ide of the hollow (from 275 km to 285 km of the lower bank of the channel) with a width of about 3 km ( Fig ure 6 1a ). The headland in the second case has the same size as the first one, but is locate d on the landward side of the hollow (from 265 km to 275 k m of the lower bank of the channel) (Fig ure 6 1 b). We would call case 1 and case 2 the seaward side case and the landward side case respectively We compare the results obtained from these two cases with th ose from the vertically homogeneous case without the headland. The depth averaged velocity ratios, the ratio of volume transport per unit width and density difference at a cross section along the central axis of the hollow at two tidal phases are shown in Fig ure 6 2 These are the times when we have obs erved the acceleration and deceleration of the flow. During flood, the velocity ratio increases linearly as the water enters the hollow and begins to have an extra increment as the flow goes across the central area of the hollow as expected. However, the c urve of the

PAGE 23

23 landward side case increases faster than the case without the headland, and the curve of the seaward side case increases slower than the straight channel case ( Fig ure 6 2a). All the three curves of the ratio of volume transport per unit width i ncrease on the seaward side of the hollow, and decreases on the landward side (Fig ure 6 2b). It means that the flow converges as the water enters the hollow, and diverges as it leaves the hollow. The curves of the density gradient oscillate along the hollo w (Fig ure 6 2c ). The variation of the density gradient difference is not consistent with the variation of the velocity ratio ( Fig ure 6 2a ), which means there are factors other than the baroclinic pressure gradient affecting the increasing of the flow veloc ity. Fig ure 6 4 shows the aerial view of the depth averaged velocity vector for the three cases during flood. We c an observe that the flow diverges from the center of the hollow to the edge of it in the seaward side of the hollow after it reaches the deepe st part of it (Fig ure 6 4 a) Moreover, we could see the flow converges as it goes into the center of the hollow, and remains converging as it leaves the hollow in the landward side case (Fig ure 6 4 b). The convergence and divergence of the flow over the hol low could be observed in the straight channel as well, even though they are not as obvious as the above two cases (Fig ure 6 4c). The convergence of the flow in the landward side case causes the higher velocity ratio, while the divergence of the flow in the seaward side case hinders the acceleration of the flow over the hollow. During ebb, the velocity ratio increases linearly as the flow enters the hollow and decreases when the water goes across the hollow. The ratio of the landward side case shows the most reduction over the hollow, followed by the ratio of the straight channel case and the seaward side case ( Fig ure 6 3a) Similar to the results during flood, the

PAGE 24

24 mass flux shows convergence and divergence of the water as the flow goes in and out of the holl ow from the landward to the seaward side (Fig ure 6 3b). The variation of the density gradient difference shows little correlation to the variation of the velocity ratio (Fig ure 6 3c) Converse to the results during flood, the flow in the seaward side case converges when the water passes the deepest part of the hollow ( Fig ure 6 5a ). The convergence of the flow induces higher velocity ratio when the flow goes across the hollow. Hence the velocity ratio in the seaward side case shows the least reduction. In th e landward side case, the divergence of the flow enhances the deceleration (Fig ure 6 5b). In conclusion, the flow accelerates as it enters the hollow during flood and decelerates during ebb due to the asymmetric distribution of the baroclinic pressure gra dient as we have shown. However, the existence of the headland changes the flow pattern by causing the convergence and divergence of the water. The headland on the upstream side of the hollow (the seaward side case during flood and the landward side case d uring ebb) makes the flow diverge when it go across the hollow. On the other hand, the headland on the downstream side of the hollow (the landward side case during flood and the seaward side case during ebb) causes convergence of the flow. The convergence/ divergence of the flow due to the location of a headland increases or decreases the flow velocity depending on the direction of the flow.

PAGE 25

25 CHAPTER 7 MOMENTUM BALANCE To explore the dynamics of the tidal flow over hollow, the momentum budget in the tidal cycle needs to be discussed .The longitudinal tidal flow is not the typical barotropic tidal wave, which consists of a balance between the barotropic pressure gradient and the local acceleration. To measure the contribution of different terms to the longi tudinal momentum balance, we calculate the averaged value of the main terms in the longitudinal momentum equation in the cross section of the deepest part of the hollow: (8.1) where refers the absolute value of any of the terms in the momentum equation and is the cross section area. To determine how the hollow affects the momentum balance, the ratio between two depth averaged terms in the longitudinal momentum equation has been calculated by: (8.2) wh ere is the momentum ratio, is the term as the numerator and is the term as the denominator represents that the is greater than the means is smaller than the means that the two terms are comparable. Now we are analyzing the vertically homogeneous case. Fig ure 8 1 shows the cross section averaged value as a function of time in one tidal cycle. The results

PAGE 26

26 indicate that the two dominant terms in the longitudinal momentum equation are the pressure gradient and the vertical friction term, which have a maximum value of The maximum value of the local acceleration term is approximately 20% smaller than them. The maximum value of the longitudinal advection ter m is about which is about 20% compared to that of the dominant terms. The lateral advection term is negligible compared to the other terms. Fig ure 8 2 shows the lowest order momentum balance in the cross section. The differenc e between the absolute value of the pressure gradient and the friction is approximately balanced by the local acceleration. The longitudinal advection term seems to be mostly smaller than the local acceleration term, although comparable at some moments. He nce the main momentum in the channel is mainly balanced by the pressure gradient, the friction and the local acceleration. The longitudinal advection term has the same order as them, and also contributes to the total momentum at some tidal phases. Fig ure 8 3 and Fig ure 8 4 show plan views of contour plots for the momentum ratio during flood and ebb. This ratio represents the relative importance of major terms in the longitudinal momentum equation and their spatial distribution. During flood, the barotropic term seems to be 1.5 to 2 times greater than the baroclinic term as the water goes to the deepest part of the hollow. However, the baroclinich term increases toward the center of the hollow and exceeds the barotropic pressure gradient on the downstream sid e ( Fig ure 8 3 a). The variation of the ratio of barotropic to baroclinic pressure gradients from the upstream side to the downstream side of the hollow indicates that the flow was mainly driven by the water surface slope as it enters the

PAGE 27

27 hollow, and then mo stly driven by the pressure gradient caused by the density gradient as it passed the center of the hollow. In other words, the proportion of the baroclinic pressure gradient in the total pressure gradient increases as the water goes across the center of th e hollow. This result is consistent with observations, which show water accelerates as it passes the deepest part of the hollow. Outside the hollow, the barotropic contribution is greater than the baroclinic contribution to the total pressuregradient. Thes e distributions are asymmetric about the longitudinal axis of the hollow but symmetric about the lateral axis of it. This asymmetric distribution in the longitudinal direction is caused by the baroclinic pressure gradient. In the hollow, the total pressure gradient is generally greater than friction, particularly in the central area of the hollow, where the water is deeper and the effects of friction would be smaller compared to the shallower area (Fig ure 8 3b). The maximum ratio between pressure and fricti on is about 1.2, on the downstream side of the hollow. This corresponds to the greater baroclinic pressure gradient there. Outside the hollow, the pressure gradient is balanced by friction, which is typical for density driven flows in estuaries. The distr ibutions shows symmetry as for the barotropic/baroclinic comparison: symmetric with respect to the lateral axis of the hollow and asymmetric relative to the longitudinal axis of it. The other two dynamical comparisons also show the longitudinal symmetry an d lateral asymmetry. The advection term is generally smaller than the pressure gradient and the friction (Fig ure 8 3c and Fig ure 8 3 d ). The general pattern of the momentum balance suggests that the flow follows density driven flow dynamics (pressure gradie nt is balanced by friction) as it enters the hollow, and the baroclinic pressure gradient is attributed to the acceleration of the flow as it moves into the hollow.

PAGE 28

28 During ebb, the barotropic pressure gradient appears comparable to the baroclinic pressure gradient before the flows goes into the hollow. However, the momentum ratio becomes smaller than 1 in the hollow, due to the larger baroclinic pressure gradient caused by the deeper depth (Fig ure 8 4 a). The ratio of the pressure gradient to the friction te rm and the pressure gradient to the longitudinal advection are generally greater than 1 in this area (Fig ure 8 4 c). It represents that the pressure gradient dominates the friction and the longitudinal advection. The ratio of the longitudinal advection and the friction is smaller than 1 outside the hollow and in the upstream area; and greater than 1 in the downstream area of the hollow (Fig ure 8 4 d). This result shows that, during ebb, friction will dominate over inertia outside the hollow; and inertia will dominate inside the hollow. It partially agrees with observations (Guo & Valle Levinson 2009). Moreover, all of the results show the typical longitudinal asymmetry/ lateral symmetry structure as we have observed during flood.

PAGE 29

29 Fig ure 2 1 A p lan view of the estuary. T h e contours represent the depth of water with an interval of 5m.

PAGE 30

30 Figure 2 2. A plan view of the hollow region. The upper panel is a panel view of the hollow region, contours represent the depth (m) with an interval o f 2 m. The lower panel shows the depth of the cross section of the longitudinal main axis of the channel. T he red dots represent the location where we obtain the velocity to calculate the velocity ratio.

PAGE 31

31 Fig ure 2 3 Model stability plot s. The blue line shows the value on the 55 th day and the red dash line shows the value on the 60 th day.

PAGE 32

32 Fig ure 3 1 Depth averaged flow velocity at the center of the hollow as a function of time. The tidal phases of acceleration and deceleration are shown.

PAGE 33

33 Fig ure 4 1 R esults of the vertically homogeneous estuary during flood. T he upper panel shows the contour plots of salinity at the cross section of the longitudinal main axis of the hollow area. The middle pan el shows the flow at the cross section of the longitudinal main axis of the hollow area. The bottom panel shows the absolute value of the depth averaged flow velocity along the channel.

PAGE 34

34 Fig ure 4 2 R esults of the vertically homogeneous estuary during ebb. T he upper panel shows the contour plots of the salinity at the cross section of the longitudinal main axis of the hollow area. The middle panel shows the flow at the cross section of the longitudinal main axis of the hollow area. The bottom panel sho ws the absolute value of the depth averaged flow velocity along the channel.

PAGE 35

35 Fig ure 4 3 R esults of the vertically stratified estuary during flood T he upper panel shows the contour plots of the salinity at the cross section of the longitudinal main ax is of the hollow area. The middle panel shows the flow at the cross section of the longitudinal main axis of the hollow area. The bottom panel shows the absolute value of the depth averaged flow velocity along the channel.

PAGE 36

36 Fig ure 4 4 R e sults of the vertically stratified estuary during ebb T he upper panel shows the contour plots of the salinity at the cross section of the longitudinal main axis of the hollow area. The middle panel shows the flow at the cross section of the longitudinal main axis of t he hollow area. The bottom panel shows the absolute value of the depth averaged flow velocity along the channel.

PAGE 37

37 Fig ure 4 5 Comparison of the results during flood. The upper panel represents the velocity ratio of the four cases along the hollow during flood. The middle panel represents the along hollow density gradient during flood. T h e bottom panel shows the bottom profile of the central axis of the hollow and the direction of the flow.

PAGE 38

38 Fig ure 4 6. Compar ison of the resu lts during ebb The upper panel represents the velocity ratio of the four cases along the hollow during ebb The middle panel represents the along hollow density gradient during flood. T h e bottom panel shows the bottom profile of the central axis of the ho llow and the direction of the flow.

PAGE 39

39 Fig ure 5 1 The velocity ratio as a function of W / L The red dotted line represents the results during flood, while the blue line is for the ebb.

PAGE 40

40 Fig ure 5 2 V elocity ratio as a functio n of The red dotted line represents the results during flood, while the blue line is for the ebb.

PAGE 41

41 (a) (b) Figure 6 1 P lan view s of two cases of the channel with headland A ) T he seaward side case B ) T he landward side case.

PAGE 42

42 Figure 6 2 Comparisons of the results from two cases during flood. A ) The depth averaged velocity rati o B ) V olume transport per unit width ratio C ) D ensity difference D ) B ottom profile at the cross section along the central a xis of the hollow during flood. Panels A ) to D ) are located from top to bottom.

PAGE 43

43 Figure 6 3. Comparisons of the results from two cases during ebb. A ) The depth averaged velocity rati o B ) V olume transport per unit width ratio C ) D ensity diffe rence D ) B ottom profile at the cross section along the central axis of the hollow during ebb Panels A ) to D ) are located from top to bottom.

PAGE 44

44 Figure 6 4 Plan view of the dept h averaged velocity vector during flood A ) T he seaward side case B ) T he landward side case C ) T he no headland case during flood. Panels A ) to C ) are from top to bottom.

PAGE 45

45 Figure 6 5. Plan view of the dept h averaged velocity vector ebb A ) T he seaward side case B ) T he landward side case C ) T he no headla nd case during ebb Panels A ) to C ) are from top to bottom.

PAGE 46

46 Fig ure 7 1 Cross section averaged momentum terms value as a function of time in one tidal cycle.

PAGE 47

47 Fig ure 7 2 Higher ordered cross section averaged momentum terms value as a functio n of time in one tidal cycle.

PAGE 48

48 Fig ure 7 3 P lan view of contour plots for the momentum ratio during flood. A ) Barotropic pressure gradient v.s. Baroclinic pressure gradient. B ) Pre ssure gradient v.s. Friction. C ) Pressure gradien t v.s. Longitudinal advection. D ) Longitudinal advection v.s Friction. The dash line shows the location of the hollow.

PAGE 49

49 Fig ure 7 4 P lan view of contour plots for the momentum ratio during ebb A ) Barotropic pressure gradient v.s. Barocli nic pressure gradient. B ) Pressure gradient v.s. Friction. C ) Pressure gradient v.s. Longitudinal advection. D ) Longitudinal advection v.s Friction. The dash line shows the location of the hollow.

PAGE 50

50 CHAPTER 8 CONCLUSION The main findings of this study are as follows: Firstly, t he vertically homogeneous case shows more pronounced flow asymmetry than the vertically stratified one. Cases with greater horizontal density gradients show greater flow accelerations/decelerations. Secondly, f or an oscillatory flow, the velocity ratio increases as W/L increases during flood and decreases as W/L decreases during ebb. Thirdly, f or an oscillatory flow, the velocity ratio increases as increases during flood and decreases as decreases during ebb. Fourthly, a headland on the upstream side of the hollow makes the flow diverge when it goes across the hollow, and the headland on the downstream side causes convergence of flow. Last but not least, t he major momentum terms in the channel are the pressure gradient, friction and local acceleration. The local acceleration is generally balanced by the difference of the pressure gradient and the friction. The longitudinal advection also contributes to the total momentum at the cent ral area of the hollow particularly during ebb.

PAGE 51

51 LIST OF REFERENCES Cheng P Valle Levinson A 200 9 Spatial v ariation of f low s tructure over e stuarine h ollows Cont inental Shelf Res earch 29 927 937. Davies, A G., Brown, J. M. 2007. Field m easuremen t and m odeling of s cour p i t d ynamics in a s andy 07 Salas Monreal D. Valle Levinson A 200 9 Continuously s tratified f low d ynamics over a h ollow Journal of Geophys ical Res earch 11 4 C0 3021 doi: 10.1029 /2007 JC00 4648 Sh chepetkin, A.F., McWilliams, J.C., 2005 The regional ocean modeling s ystem (ROMS): a split explicit, free surface, topography following coordinates ocean model Ocean Modelling 9, 347 404. Valle Levinson A Guo X. 2009 Asymmetries in t idal f low over a Seto Inland Sea s cour p it Journal of Mar ine Res earch 67 619 635

PAGE 52

52 BIOGRAPHICAL SKETCH Miao Tian received his b degree from H ohai University in 2008 He then joined the m aster s program of the University of Florida where he is currently resea rch assistant of Department of Civil & Coastal Engineering Currently, his research involves the numerical simulation of oscillatory flows over abrupt bathymetry by using ROMS. He will continue to study for his PHD degree from fall 2010